What works here is a strategy that is often helpful in construction problems: work backwards. Start with an equilateral triangle ABC and take an arbitrary line, call it a, through vertex A, and draw lines b and c parallel to a through B and C. Try rotating your picture through 60 degrees about A (so that B rotates B to C), and observe what happens to to line b during this rotation. (Warning: when working forwards from the three given parallel lines, there are generally two solution triangles; you can rotate line b clockwise and counterclockwise about the point A taken arbitrarily on line a.)

Chris